Answer :
Sure, let's analyze each sequence step by step to determine if they are arithmetic, geometric, or neither.
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- To check if it's arithmetic, we need to see if the difference between consecutive terms is constant:
[tex]\[
94.1 - 98.3 = -4.2 \\
89.9 - 94.1 = -4.2 \\
85.7 - 89.9 = -4.2
\][/tex]
Since the difference [tex]\( -4.2 \)[/tex] is constant, this is an arithmetic sequence.
2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- To check if it's arithmetic, we observe the differences between consecutive terms:
[tex]\[
0 - 1 = -1 \\
-1 - 0 = -1 \\
0 - (-1) = 1
\][/tex]
The differences are not constant, so it's not arithmetic.
- To check if it's geometric, we need to see if the ratio between consecutive terms is constant:
[tex]\[
\frac{0}{1} = 0 \quad (\text{division by zero is problematic}) \\
\frac{-1}{0} - \text{undefined}
\][/tex]
Since we encounter undefined operations, this sequence is neither arithmetic nor geometric.
3. Sequence 3: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- To check if it's arithmetic:
[tex]\[
-10.8 - (-12) = 1.2 \\
-9.6 - (-10.8) = 1.2 \\
-8.4 - (-9.6) = 1.2
\][/tex]
The difference [tex]\( 1.2 \)[/tex] is constant, so this is an arithmetic sequence.
4. Sequence 4: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- To check if it's arithmetic, look at the differences between consecutive terms:
[tex]\[
1 - (-1) = 2 \\
-1 - 1 = -2 \\
1 - (-1) = 2
\][/tex]
The differences are not constant, so it's not arithmetic.
- To check if it's geometric, look at the ratios between consecutive terms:
[tex]\[
\frac{1}{-1} = -1 \\
\frac{-1}{1} = -1 \\
\frac{1}{-1} = -1
\][/tex]
The ratio [tex]\( -1 \)[/tex] is constant, so this is a geometric sequence.
Summary:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Arithmetic
- Sequence 4: Geometric
I hope this helps! Let me know if you have any further questions.
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- To check if it's arithmetic, we need to see if the difference between consecutive terms is constant:
[tex]\[
94.1 - 98.3 = -4.2 \\
89.9 - 94.1 = -4.2 \\
85.7 - 89.9 = -4.2
\][/tex]
Since the difference [tex]\( -4.2 \)[/tex] is constant, this is an arithmetic sequence.
2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- To check if it's arithmetic, we observe the differences between consecutive terms:
[tex]\[
0 - 1 = -1 \\
-1 - 0 = -1 \\
0 - (-1) = 1
\][/tex]
The differences are not constant, so it's not arithmetic.
- To check if it's geometric, we need to see if the ratio between consecutive terms is constant:
[tex]\[
\frac{0}{1} = 0 \quad (\text{division by zero is problematic}) \\
\frac{-1}{0} - \text{undefined}
\][/tex]
Since we encounter undefined operations, this sequence is neither arithmetic nor geometric.
3. Sequence 3: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- To check if it's arithmetic:
[tex]\[
-10.8 - (-12) = 1.2 \\
-9.6 - (-10.8) = 1.2 \\
-8.4 - (-9.6) = 1.2
\][/tex]
The difference [tex]\( 1.2 \)[/tex] is constant, so this is an arithmetic sequence.
4. Sequence 4: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- To check if it's arithmetic, look at the differences between consecutive terms:
[tex]\[
1 - (-1) = 2 \\
-1 - 1 = -2 \\
1 - (-1) = 2
\][/tex]
The differences are not constant, so it's not arithmetic.
- To check if it's geometric, look at the ratios between consecutive terms:
[tex]\[
\frac{1}{-1} = -1 \\
\frac{-1}{1} = -1 \\
\frac{1}{-1} = -1
\][/tex]
The ratio [tex]\( -1 \)[/tex] is constant, so this is a geometric sequence.
Summary:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Arithmetic
- Sequence 4: Geometric
I hope this helps! Let me know if you have any further questions.
